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2.    (a)   Let E, F be two metrizable locally compact spaces and let TT: E->F be a con-
tinuous map. The mapping TT is said to be proper if, for each compact subset K of F,
the inverse image TT'^K) is compact. Show that, if TT is proper and y e 7r(E), every
neighborhood in E of the set 7r~l(y) contains a neighborhood of the form TT'^U),
where U is a neighborhood of y in F.

(b)    Let g be a lower semicontinuous real-valued function on E. For each y e F,
let/0>) be the greatest lower bound of g on the set rr~l(y) (so that/(y) = +00 if
rr~l(y) = 0). Show that /is lower semicontinuous on F.

(c)    Let g be the continuous function (xt, x2)t->\XiX2 — 1| on R x ]0, -f oo [. For
each x e R, let/(*i) = inf g(xi9 x2). Show that/is not lower semicontinuous onR.


3.    For each point / = (tl9 . . . , tn) e C", put P,(X) = X" + t^"1 + ••• + /„. Show that
there exists a continuous mapping t\-^x(t) of C" into R and a lower semicontinuous
mapping t \-+y(t) of C" into R, such that Pf(x(t) + iy(t)) = 0 for all t e C1. (Take
x(t) to be the largest of the real parts of the roots of P, , and use (9.1 7.4)).

4.    Let / be any mapping of a metric space E into a metric space F. For each x e E, let
Łl(x) be the oscillation of /at x with respect to E (3.14), so that Łl(x) is a positive real
number or + °° • Show that the mapping x \— > Q(x) of E into R is upper semicontinuous.

5.    Let E be a metric space and let/: E ->R be lower semicontinuous. Let <re E be a point
at which the oscillation Q,(a) of / is finite. Show that for each e > 0, there exists a
neighborhood V of a in E such that inf Łl(x) fŁ e. (Show that otherwise there would

X 6 V

exist points x arbitrarily close to a at which /(#) took arbitrarily large values.)

6.    Let (fl,,)oi be an infinite sequence of distinct points in the interval [0, 1[. For each
integer N>1, let bi, ..., b^ denote the sequence obtained by arranging the set
{fli, ...,0N} in increasing order. The intervals [0, ŁJ, [bi9b2[9...9 [6N-i, ŁN[»[^N, H
are said to form the NT//z subdivision of [0, 1 [ corresponding to the sequence (an).
Let WN (resp. v^) denote the minimum (resp. maximum) of the lengths of the intervals
of the Nth subdivision. Let A = lim inf NJ/N , JJL = lim sup NuN .

N-+OO                             N-+OO

(a) For each e > 0, let n0 be such that #N ;> (A - e)/N and yN <; (/* + e)/N for all
N ^ /?0 • Show that, for each N ^> «0 and each integer r such that 0 ^ r ^ N» there
exist 2r intervals of the (N + r)th subdivision whose lengths are greater than or equal
to the 2r numbers

X — s     A — e     A — e      A — e         A — e     A — e

NTT'   N + 2'   N + 2''N-f-rf   N+7

and such that the other intervals of the (N + r)th subdivision are intervals of the Nth
subdivision (use induction on r).

(b)   Show that, if N §: nQ , the intervals of the Nth subdivision can be arranged in
order in such a way that their lengths are less than or equal to the N + 1 numbers


N   '    N.+ 1''"'   2N  '

(For each interval I of the Nth subdivision, let s(l) be the largest integer ^2N such
that I is an interval of the s(I)th subdivision, and arrange the intervals I in order of
increasing .s(I).)(cf. (3.13.3) and (3.13.4)). To get corresponding results in general, it is